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Let $a_1,\\dots,a_k,b_k,\\dots,b_1$ be vertices placed in a counterclockwise order on the outer face of $G$. We show that the $k\\times k$ matrix of the two-point spin correlation functions \\[\n  M_{i,j} = \\langle \\sigma_{a_i} \\sigma_{b_j} \\rangle\n  \\] is totally nonnegative. Moreover, $\\det M > 0$ if and only if there exist $k$ pairwise vertex-disjoint paths that connect $a_i$ with $b"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.06068","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-20T11:36:01Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"196a42c76cb471b64900554111afed68ed972781e483056b1af8ed3a771b98d3","abstract_canon_sha256":"c625f7f80ac8008bf1d52ea56727986631046f5330bac9d2b2031ab09e219aeb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:50.888118Z","signature_b64":"YoM4yX1LYAMBP2uqnewM3QY0QKFeeyp4G7EShcdNIY5P2qWVTWPDuDOJ6Yzy/nvy9WWw3I3cTP8rbcLrT2w6CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4c43ed232c9fbe2a9b7ba17775271e9101269d99efac0cdbf30f7387897b1cd","last_reissued_at":"2026-05-18T00:53:50.887759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:50.887759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The planar Ising model and total positivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.PR","authors_text":"Marcin Lis","submitted_at":"2016-06-20T11:36:01Z","abstract_excerpt":"A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph $G$. Let $a_1,\\dots,a_k,b_k,\\dots,b_1$ be vertices placed in a counterclockwise order on the outer face of $G$. We show that the $k\\times k$ matrix of the two-point spin correlation functions \\[\n  M_{i,j} = \\langle \\sigma_{a_i} \\sigma_{b_j} \\rangle\n  \\] is totally nonnegative. Moreover, $\\det M > 0$ if and only if there exist $k$ pairwise vertex-disjoint paths that connect $a_i$ with $b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06068","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.06068","created_at":"2026-05-18T00:53:50.887820+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.06068v3","created_at":"2026-05-18T00:53:50.887820+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.06068","created_at":"2026-05-18T00:53:50.887820+00:00"},{"alias_kind":"pith_short_12","alias_value":"6TCD5URSZH56","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"6TCD5URSZH56FKNX","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"6TCD5URS","created_at":"2026-05-18T12:30:04.600751+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E","json":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E.json","graph_json":"https://pith.science/api/pith-number/6TCD5URSZH56FKNXXILXOUTR5E/graph.json","events_json":"https://pith.science/api/pith-number/6TCD5URSZH56FKNXXILXOUTR5E/events.json","paper":"https://pith.science/paper/6TCD5URS"},"agent_actions":{"view_html":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E","download_json":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E.json","view_paper":"https://pith.science/paper/6TCD5URS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.06068&json=true","fetch_graph":"https://pith.science/api/pith-number/6TCD5URSZH56FKNXXILXOUTR5E/graph.json","fetch_events":"https://pith.science/api/pith-number/6TCD5URSZH56FKNXXILXOUTR5E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E/action/storage_attestation","attest_author":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E/action/author_attestation","sign_citation":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E/action/citation_signature","submit_replication":"https://pith.science/pith/6TCD5URSZH56FKNXXILXOUTR5E/action/replication_record"}},"created_at":"2026-05-18T00:53:50.887820+00:00","updated_at":"2026-05-18T00:53:50.887820+00:00"}